Cubicity, boxicity, and vertex cover

A k-dimensional box is the cartesian product R-1 x R-2 x ... x R-k where each R-i is a closed interval on the real line. The boxicity of a graph G,denoted as box(G), is the minimum integer k such that G is the intersection graph of a collection of k-dimensional boxes. A unit cube in k-dimensional space or a k-cube is defined as the cartesian product R-1 x R-2 x ... x R-k where each Ri is a closed interval on the real line of the form [a(i), a(i) + 1]. The cubicity of G, denoted as cub(G), is the minimum k such that G is the intersection graph of a collection of k-cubes. In this paper we show that cub(G) <= t + inverted right perpendicularlog(n - t)inverted left perpendicular - 1 and box(G) <= left perpendiculart/2right perpendicular + 1, where t is the cardinality of a minimum vertex cover of G and n is the number of vertices of G. We also show the tightness of these upper bounds. F.S. Roberts in his pioneering paper on boxicity and cubicity had shown that for a graph G, box(G) <= left perpendicularn/2right perpendicular and cub(G) <= inverted right perpendicular2n/3inverted left perpendicular, where n is the number of vertices of G, and these bounds are tight. We show that if G is a bipartite graph then box(G) <= inverted right perpendicularn/4inverted left perpendicular and this bound is tight. We also show that if G is a bipartite graph then cub(G) <= n/2 + inverted right perpendicularlog n inverted left perpendicular - 1. We point out that there exist graphs of very high boxicity but with very low chromatic number. For example there exist bipartite (i.e., 2 colorable) graphs with boxicity equal to n/4. Interestingly, if boxicity is very close to n/2, then chromatic number also has to be very high. In particular, we show that if box(G) = n/2 - s, s >= 0, then chi (G) >= n/2s+2, where chi (G) is the chromatic number of G.

On the Cubicity of Interval Graphs

A k-cube (or ``a unit cube in k dimensions'') is defined as the Cartesian product R-1 x . . . x R-k where R-i (for 1 <= i <= k) is an interval of the form [a(i), a(i) + 1] on the real line. The k-cube representation of a graph G is a mapping of the vertices of G to k-cubes such that the k-cubes corresponding to two vertices in G have a non-empty intersection if and only if the vertices are adjacent. The cubicity of a graph G, denoted as cub(G), is defined as the minimum dimension k such that G has a k-cube representation. An interval graph is a graph that can be represented as the intersection of intervals on the real line - i. e., the vertices of an interval graph can be mapped to intervals on the real line such that two vertices are adjacent if and only if their corresponding intervals overlap. We show that for any interval graph G with maximum degree Delta, cub(G) <= inverted right perpendicular log(2) Delta inverted left perpendicular + 4. This upper bound is shown to be tight up to an additive constant of 4 by demonstrating interval graphs for which cubicity is equal to inverted right perpendicular log(2) Delta inverted left perpendicular.

A quantum chemical investigation of the metal centres in cytochrome c oxidase

Quantum effects are often of key importance for the function of biological systems at molecular level. Cellular respiration, where energy is extracted from the reduction of molecular oxygen to water, is no exception. In this work, the end station of the electron transport chain in mitochondria, cytochrome c oxidase, is investigated using quantum chemical methodology. Cytochrome c oxidase contains two haems, haem a and haem a3. Haem a3, with its copper companion, CuB, is involved in the final reduction of oxygen into water. This binuclear centre receives the necessary electrons from haem a. Haem a, in turn, receives its electrons from a copper ion pair in the vicinity, called CuA. Density functional theory (DFT) has been used to clarify the charge and spin distributions of haem a, as well as changes in these during redox activity. Upon reduction, the added electron is shown to be evenly distributed over the entire haem structure, important for the accommodation of the prosthetic group within the protein. At the same time, the spin distribution of the open-shell oxidised state is more localised to the central iron. The exact spin density distribution has been disputed in the literature, however, different experiments indicating different distributions of the unpaired electron. The apparent contradiction is shown to be due to the false assumption of a unit amount of unpaired electron density; in fact, the oxidised state has about 1.3 unpaired electrons. The validity of the DFT results have been corroborated by wave function based coupled cluster calculations. Point charges, for use in classical force field based simulations, have been parameterised for the four metal centres, using a newly developed methodology. In the procedure, the subsystem for which point charges are to be obtained, is surrounded by an outer region, with the purpose of stabilising the inner region, both electronically and structurally. Finally, the possibility of vibrational promotion of the electron transfer step between haem a and a3 has been investigated. Calculating the full vibrational spectra, at DFT level, of a combined model of the two haems, revealed several normal modes that do shift electron density between the haems. The magnitude of the shift was found to be moderate, at most. The proposed mechanism could have an assisting role in the electron transfer, which still seems to be dominated by electron tunnelling.

Elevated CDCP1 predicts poor patient outcome and mediates ovarian clear cell carcinoma by promoting tumor spheroid formation, cell migration and chemoresistance

Hematogenous metastases are rarely present at diagnosis of ovarian clear cell carcinoma (OCC). Instead dissemination of these tumors is characteristically via direct extension of the primary tumor into nearby organs and the spread of exfoliated tumor cells throughout the peritoneum, initially via the peritoneal fluid, and later via ascites that accumulates as a result of disruption of the lymphatic system. The molecular mechanisms orchestrating these processes are uncertain. In particular, the signaling pathways used by malignant cells to survive the stresses of anchorage-free growth in peritoneal fluid and ascites, and to colonize remote sites, are poorly defined. We demonstrate that the transmembrane glycoprotein CUB-domain-containing protein 1 (CDCP1) has important and inhibitable roles in these processes. In vitro assays indicate that CDCP1 mediates formation and survival of OCC spheroids, as well as cell migration and chemoresistance. Disruption of CDCP1 via silencing and antibody-mediated inhibition markedly reduce the ability of TOV21G OCC cells to form intraperitoneal tumors and induce accumulation of ascites in mice. Mechanistically our data suggest that CDCP1 effects are mediated via a novel mechanism of protein kinase B (Akt) activation. Immunohistochemical analysis also suggested that CDCP1 is functionally important in OCC, with its expression elevated in 90% of 198 OCC tumors and increased CDCP1 expression correlating with poor patient disease-free and overall survival. This analysis also showed that CDCP1 is largely restricted to the surface of malignant cells where it is accessible to therapeutic antibodies. Importantly, antibody-mediated blockade of CDCP1 in vivo significantly increased the anti-tumor efficacy of carboplatin, the chemotherapy most commonly used to treat OCC. In summary, our data indicate that CDCP1 is important in the progression of OCC and that targeting pathways mediated by this protein may be useful for the management of OCC, potentially in combination with chemotherapies and agents targeting the Akt pathway.

On the Cubicity of AT-Free Graphs and Circular-Arc Graphs

A unit cube in k dimensions (k-cube) is defined as the Cartesian product R-1 x R-2 x ... x R-k where R-i (for 1 <= i <= k) is a closed interval of the form [a(i), a(i) + 1] on the real line. A graph G on n nodes is said to be representable as the intersection of k-cubes (cube representation in k dimensions) if each vertex of C can be mapped to a k-cube such that two vertices are adjacent in G if and only if their corresponding k-cubes have a non-empty intersection. The cubicity of G denoted as cub(G) is the minimum k for which G can be represented as the intersection of k-cubes. An interesting aspect about cubicity is that many problems known to be NP-complete for general graphs have polynomial time deterministic algorithms or have good approximation ratios in graphs of low cubicity. In most of these algorithms, computing a low dimensional cube representation of the given graph is usually the first step. We give an O(bw . n) algorithm to compute the cube representation of a general graph G in bw + 1 dimensions given a bandwidth ordering of the vertices of G, where bw is the bandwidth of G. As a consequence, we get O(Delta) upper bounds on the cubicity of many well-known graph classes such as AT-free graphs, circular-arc graphs and cocomparability graphs which have O(Delta) bandwidth. Thus we have: 1. cub(G) <= 3 Delta - 1, if G is an AT-free graph. 2. cub(G) <= 2 Delta + 1, if G is a circular-arc graph. 3. cub(G) <= 2 Delta, if G is a cocomparability graph. Also for these graph classes, there axe constant factor approximation algorithms for bandwidth computation that generate orderings of vertices with O(Delta) width. We can thus generate the cube representation of such graphs in O(Delta) dimensions in polynomial time.

Boxicity and cubicity of asteroidal triple free graphs

The boxicity of a graph G, denoted box(G), is the least integer d such that G is the intersection graph of a family of d-dimensional (axis-parallel) boxes. The cubicity, denoted cub(G), is the least dsuch that G is the intersection graph of a family of d-dimensional unit cubes. An independent set of three vertices is an asteroidal triple if any two are joined by a path avoiding the neighbourhood of the third. A graph is asteroidal triple free (AT-free) if it has no asteroidal triple. The claw number psi(G) is the number of edges in the largest star that is an induced subgraph of G. For an AT-free graph G with chromatic number chi(G) and claw number psi(G), we show that box(G) <= chi(C) and that this bound is sharp. We also show that cub(G) <= box(G)([log(2) psi(G)] + 2) <= chi(G)([log(2) psi(G)] + 2). If G is an AT-free graph having girth at least 5, then box(G) <= 2, and therefore cub(G) <= 2 [log(2) psi(G)] + 4. (c) 2010 Elsevier B.V. All rights reserved.

The hardness of approximating the boxicity, cubicity and threshold dimension of a graph

A k-dimensional box is the Cartesian product R-1 X R-2 X ... X R-k where each R-i is a closed interval on the real line. The boxicity of a graph G, denoted as box(G), is the minimum integer k such that G can be represented as the intersection graph of a collection of k-dimensional boxes. A unit cube in k-dimensional space or a k-cube is defined as the Cartesian product R-1 X R-2 X ... X R-k where each R-i is a closed interval oil the real line of the form a(i), a(i) + 1]. The cubicity of G, denoted as cub(G), is the minimum integer k such that G can be represented as the intersection graph of a collection of k-cubes. The threshold dimension of a graph G(V, E) is the smallest integer k such that E can be covered by k threshold spanning subgraphs of G. In this paper we will show that there exists no polynomial-time algorithm for approximating the threshold dimension of a graph on n vertices with a factor of O(n(0.5-epsilon)) for any epsilon > 0 unless NP = ZPP. From this result we will show that there exists no polynomial-time algorithm for approximating the boxicity and the cubicity of a graph on n vertices with factor O(n(0.5-epsilon)) for any epsilon > 0 unless NP = ZPP. In fact all these hardness results hold even for a highly structured class of graphs, namely the split graphs. We will also show that it is NP-complete to determine whether a given split graph has boxicity at most 3. (C) 2010 Elsevier B.V. All rights reserved.

Cubicity of Interval Graphs and the Claw Number

Let G(V, E) be a simple, undirected graph where V is the set of vertices and E is the set of edges. A b-dimensional cube is a Cartesian product l(1) x l(2) x ... x l(b), where each l(i) is a closed interval of unit length on the real line. The cub/city of G, denoted by cub(G), is the minimum positive integer b such that the vertices in G can be mapped to axis parallel b-dimensional cubes in such a way that two vertices are adjacent in G if and only if their assigned cubes intersect. An interval graph is a graph that can be represented as the intersection of intervals on the real line-i.e. the vertices of an interval graph can be mapped to intervals on the real line such that two vertices are adjacent if and only if their corresponding intervals overlap. Suppose S(m) denotes a star graph on m+1 nodes. We define claw number psi(G) of the graph to be the largest positive integer m such that S(m) is an induced subgraph of G. It can be easily shown that the cubicity of any graph is at least log(2) psi(G)]. In this article, we show that for an interval graph G log(2) psi(G)-]<= cub(G)<=log(2) psi(G)]+2. It is not clear whether the upper bound of log(2) psi(G)]+2 is tight: till now we are unable to find any interval graph with cub(G)> (log(2)psi(G)]. We also show that for an interval graph G, cub(G) <= log(2) alpha], where alpha is the independence number of G. Therefore, in the special case of psi(G)=alpha, cub(G) is exactly log(2) alpha(2)]. The concept of cubicity can be generalized by considering boxes instead of cubes. A b-dimensional box is a Cartesian product l(1) x l(2) x ... x l(b), where each I is a closed interval on the real line. The boxicity of a graph, denoted box(G), is the minimum k such that G is the intersection graph of k-dimensional boxes. It is clear that box(G)<= cub(G). From the above result, it follows that for any graph G, cub(G) <= box(G)log(2) alpha]. (C) 2010 Wiley Periodicals, Inc. J Graph Theory 65: 323-333, 2010

Playing Games on Sets and Models

The most prominent objective of the thesis is the development of the generalized descriptive set theory, as we call it. There, we study the space of all functions from a fixed uncountable cardinal to itself, or to a finite set of size two. These correspond to generalized notions of the universal Baire space (functions from natural numbers to themselves with the product topology) and the Cantor space (functions from natural numbers to the {0,1}-set) respectively. We generalize the notion of Borel sets in three different ways and study the corresponding Borel structures with the aims of generalizing classical theorems of descriptive set theory or providing counter examples. In particular we are interested in equivalence relations on these spaces and their Borel reducibility to each other. The last chapter shows, using game-theoretic techniques, that the order of Borel equivalence relations under Borel reduciblity has very high complexity. The techniques in the above described set theoretical side of the thesis include forcing, general topological notions such as meager sets and combinatorial games of infinite length. By coding uncountable models to functions, we are able to apply the understanding of the generalized descriptive set theory to the model theory of uncountable models. The links between the theorems of model theory (including Shelah's classification theory) and the theorems in pure set theory are provided using game theoretic techniques from Ehrenfeucht-Fraïssé games in model theory to cub-games in set theory. The bottom line of the research declairs that the descriptive (set theoretic) complexity of an isomorphism relation of a first-order definable model class goes in synch with the stability theoretical complexity of the corresponding first-order theory. The first chapter of the thesis has slightly different focus and is purely concerned with a certain modification of the well known Ehrenfeucht-Fraïssé games. There we (me and my supervisor Tapani Hyttinen) answer some natural questions about that game mainly concerning determinacy and its relation to the standard EF-game

Structural model for the Cu-B site of dopamine beta-hydroxylase: Crystal structure of a copper(II) complex showing N3OS coordination with an axial sulfur ligation

A copper(II) complex containing a NSO-donor Schiff base and NN-donor 2,2'-bipyridine has been prepared and structurally characterized. The square pyramidal complex with an axial sulfur ligation is a structural model for the CUB site of dopamine-hydroxylase in its oxidized form. The copper(II) complex is catalytically active in the oxidation of ascorbic acid by dioxygen mediated by a copper(I) species which is proposed to have a four-coordinate structure with a N3S coordination geometry.

Polynomial time and parameterized approximation algorithms for boxicity

The boxicity (cubicity) of a graph G, denoted by box(G) (respectively cub(G)), is the minimum integer k such that G can be represented as the intersection graph of axis parallel boxes (cubes) in ℝ k . The problem of computing boxicity (cubicity) is known to be inapproximable in polynomial time even for graph classes like bipartite, co-bipartite and split graphs, within an O(n 0.5 − ε ) factor for any ε > 0, unless NP = ZPP. We prove that if a graph G on n vertices has a clique on n − k vertices, then box(G) can be computed in time n22O(k2logk) . Using this fact, various FPT approximation algorithms for boxicity are derived. The parameter used is the vertex (or edge) edit distance of the input graph from certain graph families of bounded boxicity - like interval graphs and planar graphs. Using the same fact, we also derive an O(nloglogn√logn√) factor approximation algorithm for computing boxicity, which, to our knowledge, is the first o(n) factor approximation algorithm for the problem. We also present an FPT approximation algorithm for computing the cubicity of graphs, with vertex cover number as the parameter.

Upper bound on cubicity in terms of boxicity for graphs of low chromatic number

The boxicity (respectively cubicity) of a graph G is the least integer k such that G can be represented as an intersection graph of axis-parallel k-dimensional boxes (respectively k-dimensional unit cubes) and is denoted by box(G) (respectively cub(G)). It was shown by Adiga and Chandran (2010) that for any graph G, cub(G) <= box(G) log(2) alpha(G], where alpha(G) is the maximum size of an independent set in G. In this note we show that cub(G) <= 2 log(2) X (G)] box(G) + X (G) log(2) alpha(G)], where x (G) is the chromatic number of G. This result can provide a much better upper bound than that of Adiga and Chandran for graph classes with bounded chromatic number. For example, for bipartite graphs we obtain cub(G) <= 2(box(G) + log(2) alpha(G)] Moreover, we show that for every positive integer k, there exist graphs with chromatic number k such that for every epsilon > 0, the value given by our upper bound is at most (1 + epsilon) times their cubicity. Thus, our upper bound is almost tight. (c) 2015 Elsevier B.V. All rights reserved.

Visualization Summit 2007: Ten research goals for 2010

At the first international Visualization Summit, more than 100 international researchers and practitioners defined and assessed nine original and important research goals in the context of Visualization Science, and proposed methods for achieving these goals by 2010. The synthesis of the whole event is presented in the 10th research goal. This article contributes a building block for systemizing visualization research by proposing mutually elaborated research goals with defined milestones. Such a consensus on where to go together is only one step toward establishing visualization science in the long-term perspective as a discipline with comparable relevance to chemistry, mathematics, language, or history. First, this article introduces the conference setting. Second, it describes the research goals and findings from the nine workshops. Third, a survey among 62 participants about the originality and importance of each research goal is presented and discussed. Finally, the article presents a synthesis of the nine research goals in the form of a 10th research goal, namely Visualizing Future Cities. The article is relevant for visualization researchers, trend scouts, research programme directors who define the topics that get funds. © 2007 Palgr aveMacmillan Ltd. All rights reserved.